Our seminar will meet on a biweekly schedule, on Wednesdays, starting January 17, 2018. All the mathematicians in the Atlanta area, at all stages in their career are welcome to participate! The talks are intended for a general audience.

Abstract: Quadratic fields arise naturally in the study of primes and in solving Diophantine equations. Their arithmetic is in many ways similar to the familiar arithmetic of whole numbers, but an important difference is that in quadratic fields, integers cannot be factored uniquely into primes. Class numbers measure the degree to which unique factorization fails, and it turns out that consequences of deep knowledge about these numbers percolate through virtually every important question in number theory. We’ll look at some of the rich history in the study of class numbers and at important questions that are the subject of current research.

Abstract: We provide a multiple time scales analysis for the Swift-Hohenberg equation with delayed feedback via Pyragas control, focusing on Turing bifurcations with and without additive noise. In the deterministic case, a Ginzburg-Landau-type modulation equation is derived that inherits Pyragas control terms from the original equation. The Eckhaus stability criteria is obtained for the rolls, with the delay driving the appearance of an intermediate time scale observed in the transients. In the stochastic context, slow modulation equations are derived for the amplitudes of the primary modes that are coupled to a fast Ornstein-Uhlenbeck-type equation with delay for the zero mode driven by the additive noise. By deriving an averaging approximation for the amplitude of the primary mode, we show how the interaction of noise and delay influences the existence and stability range for the noisy roll-type patterns. Furthermore, approximations for the spectral densities of the primary and zero modes show that oscillations on the intermediate times scale are sustained through the phenomenon of coherence resonance. These dynamics on the intermediate time scale are sustained through the interaction of noise and delay, in contrast to the deterministic context where dynamics on the intermediate times scale are transient.

Abstract: In this talk, we will outline a new approach to the development of iteratively regularized Broyden-type algorithms for solving nonlinear ill-posed inverse problems in either finite or infinite dimensional spaces. The novel regularization methods are designed to solve large-scale unstable least squares problems, where the Jacobian of a discretized nonlinear operator is difficult or even infeasible to compute. To face this challenge, a family of Gauss-Newton and Levenberg-Marquardt algorithms with the Fréchet derivative operator recalculated iteratively by using Broyden-type single rank updates is considered. To balance accuracy and stability, the pseudo-inverse for the derivative-free Jacobian is regularized in a problem-specific manner at every step of the iteration process. The study has been motivated by applications of inverse problems in epidemiology and infectious disease modeling, where stable estimation of key epidemiological parameters at the onset of an emerging virus is paramount in assisting public health authorities to rapidly assess the situation in order to determine whether the pathogen in question is capable of generating sustained local or global outbreaks. To illustrate theoretical findings, numerical simulations for both parameter estimation and forecasting of future incidence cases will be presented.

Abstract: The estimation of singular integrals, which are generally non-local and non-positive operators, by sparse forms, which by contrast are positive and localized, has recently become a leading trend in Calderon-Zygmund theory and beyond. In this talk we will explore the concept of sparse domination and its implications for weighted norm inequalities. In particular, we will discuss several sparse domination results, including the study of the bilinear Hilbert transform and rough homogeneous singular integrals.